##
**Commutativity for a certain class of rings.**
*(English)*
Zbl 0839.16029

For rings \(R\), the author considers four similar conditions involving commutators, of which the first is \((c_1)\): “For every \(x, y\in R\) there holds \(x^r[x^s, y]= \pm[ x, y^t] x^n\) with integers \(t> 1\), \(s\geq 1\), \(n\geq 0\), \(r\geq 0\).” His theorems assert that a ring \(R\) with 1 must be commutative if it satisfies one of these conditions in conjunction with an appropriate restriction on additive torsion. Apparently \(t\), \(s\), \(n\), and \(r\) are fixed, but it is not clear how to interpret the ambiguity in sign in condition \((c_1)\) and the other three conditions. The safe interpretation is that one chooses the same sign for all \(x, y\in R\).

While the theorems of this paper are similar to many known results, the proofs do have an element of novelty; they provide an interesting application of some earlier results of T. P. Kezlan [Math. Jap. 29, 135-139 (1984; Zbl 0538.16028)].

While the theorems of this paper are similar to many known results, the proofs do have an element of novelty; they provide an interesting application of some earlier results of T. P. Kezlan [Math. Jap. 29, 135-139 (1984; Zbl 0538.16028)].

Reviewer: H.E.Bell (St.Catherines)

### MSC:

16U70 | Center, normalizer (invariant elements) (associative rings and algebras) |

16U80 | Generalizations of commutativity (associative rings and algebras) |

16R50 | Other kinds of identities (generalized polynomial, rational, involution) |

### Citations:

Zbl 0538.16028
PDF
BibTeX
XML
Cite

\textit{H. A. S. Abujabal}, Georgian Math. J. 3, No. 1, 1--10 (1996; Zbl 0839.16029)

### References:

[1] | H. A. S. Abujabal and V. Perić, Commutativity ofs-unital rings through a Streb result.Radovi Mat. 7 (1991), 73–92. · Zbl 0770.16015 |

[2] | H. Komatsu, A commutativity theorem for rings.Math. J. Okayama Univ. 26 (1984), 109–111. · Zbl 0568.16017 |

[3] | M. A. Quadri and M. A. Khan, A commutativity theorem for associative rings.Math. Japonica 33 (1988), 275–279. · Zbl 0655.16021 |

[4] | H. G. Moore, Generalizedn-like rings and commutativity.Canad. Math. Bull. 23 (1980), 449–452. · Zbl 0447.16027 |

[5] | N. Jacobson, Structure of rings.Amer. Math. Soc., Colloq. Publ., 1964. |

[6] | W. K. Nicholson and A. Yaqub, Commutativity theorems for rings and groups.Canad. Math. Bull. 22 (1979), 419–423. · Zbl 0605.16020 |

[7] | T. P. Kezlan, A note on commutativity of semi-prime PI rings.Math. Japonica 27 (1982), 267–268. · Zbl 0481.16013 |

[8] | T. P. Kezlan, On identities which are equivalent with commutativity.Math. Japonica 29 (1984), 135–139. · Zbl 0538.16028 |

[9] | H. E. Bell, M. A. Quadri, and M. Ashraf Commutativity of rings with some commutator constraints.Radovi Mat. 5 (1989), 223–230. · Zbl 0697.16031 |

[10] | T. P. Kezlan, On commutativity theorems for PI ring with unity.Tamkang J. Math. 24 (1993), 29–36. · Zbl 0810.16033 |

[11] | H. E. Bell, M. A. Quadri, and M. A. Khan, Two commutativity theorems for rings.Radovi Mat. 3 (1987), 255–260. · Zbl 0648.16028 |

[12] | Y. Hirano, Y. Kobayashi, and H. Tominaga, Some polynomial identities and commutativity ofs-unital rings.Math. J. Okayama Univ. 24 (1982), 7–13. · Zbl 0487.16023 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.